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Abstract

In this paper, we are concerned with a system of nonlinear viscoelastic wave equations
with initial and Dirichlet boundary conditions in (). Under suitable assumptions, we establish a general decay result by multiplier techniques,
which extends some existing results for a single equation to the case of a coupled
system.

MSC:
35L05, 35L55, 35L70.

Keywords:

viscoelastic system; general decay; weak damping

1 Introduction

In this paper, we are concerned with a coupled system of nonlinear viscoelastic wave
equations with weak damping

(1.1)

where () is a bounded domain with smooth boundary ∂Ω, u and v represent the transverse displacements of waves. The functions and denote the kernel of a memory, and are the nonlinearities.

In recent years, many mathematicians have paid their attention to the energy decay
and dynamic systems of the nonlinear wave equations, hyperbolic systems and viscoelastic
equations.

Firstly, we recall some results concerning single viscoelastic wave equation. Kafini
and Tatar [1] considered the following Cauchy problem:

(1.2)

They established the polynomial decay of the first-order energy of solutions for
compactly supported initial data and for a not necessarily decreasing relaxation function.
Later Tatar [2] studied the problem (1.2) with the Dirichlet boundary condition and showed that the
decay of solutions was an arbitrary decay not necessarily at exponential or polynomial
rate. Cavalcanti et al.[3] studied the following equation with Dirichlet boundary condition:

The authors established a global existence result for and an exponential decay of energy for . They studied the interaction within the and the memory term . Later on, several other results were published based on [4-6]. For more results on a single viscoelastic equation, we can refer to [7-14].

For a coupled system, Agre and Rammaha [15] investigated the following system:

where () is a bounded domain with smooth boundary. They considered the following assumptions
on ():

(A1) Let

with , if and if ; .

(A2) There exist two positive constants , such that for all , satisfies

Under the assumptions (A1)-(A2), they established the global existence of weak solutions and the global existence
of small weak solutions with initial and Dirichlet boundary conditions. Moreover,
they also obtained the blow up of weak solutions. Mustafa [16] studied the following system:

(1.3)

in with initial and Dirichlet boundary conditions, proved the existence and uniqueness
to the system by using the classical Faedo-Galerkin method and established a stability
result by multiplier techniques. But the author considered the following different
assumptions on () from (A1)-(A2):

() () are functions and there exists a function F such that

()

for all , where the constant and , for .

Han and Wang [17] considered the following coupled nonlinear viscoelastic wave equations with weak
damping:

(1.4)

where is a bounded domain with smooth boundary ∂Ω. Under the assumptions (A1)-(A2) on (), the initial data and the parameters in the equations, they established the local
existence, global existence uniqueness and finite time blow up properties. When the
weak damping terms , were replaced by the strong damping terms , , Liang and Gao [18] showed that under certain assumption on initial data in the stable set, the decay
rate of the solution energy is exponential when they take

and if , if . Moreover, they obtained that the solutions with positive initial energy blow up
in a finite time for certain initial data in the unstable set. For more results on
coupled viscoelastic equations, we can refer to [19-21].

If we take in (1.4), the system will be transformed into (1.1). To the best of our knowledge,
there is no result on general energy decay for the viscoelastic problem (1.1). Motivated
by [16,17], in this paper, we shall establish the general energy decay for the problem (1.1)
by multiplier techniques, which extends some existing results for a single equation
to the case of a coupled system. The rest of our paper is organized as follows. In
Section 2, we give some preparations for our consideration and our main result. The
statement and the proof of our main result will be given in Section 3.

For the reader’s convenience, we denote the norm and the scalar product in by and , respectively. denotes a general constant, which may be different in different estimates.

2 Preliminaries and main result

To state our main result, in addition to (A1)-(A2), we need the following assumption.

(A3) , , are differentiable functions such that

and there exist nonincreasing functions satisfying

Now, we define the energy functional

(2.1)

and the functional

(2.2)

where

The existence of a global solution to the system (1.1) is established in [17] as follows.

3 Proof of Theorem 2.1

In this section, we carry out the proof of Theorem 2.1. Firstly, we will estimate
several lemmas.

Lemma 3.1Let, be the solution of (1.1). Then the following energy estimate holds for any:

(3.1)

Proof Multiplying the first equation of (1.1) by and the second equation by , respectively, integrating the results over Ω, performing integration by parts and
noting that , we can easily get (3.1). The proof is complete. □

Lemma 3.2Under the assumption (A3), the following hold:

(3.2)

(3.3)

Proof Using Hölder’s inequality, we get

On the other hand, we repeat the above proof with , instead of g, we can get (3.3). The proof is now complete. □

Lemma 3.3Let (A1)-(A3) hold and, be the solution of (1.1). Then the functionaldefined by

satisfies

(3.4)

for all.

Proof By (1.1), a direct differentiation gives

(3.5)

From the assumptions (A1)-(A2), we derive

and

(3.6)

By Young’s inequality and (3.2), we deduce for any

(3.7)

Similarly, we have

(3.8)

Using Young’s inequality and Poincaré’s inequality, we obtain for any

(3.9)

where λ is the first eigenvalue of −Δ with the Dirichlet boundary condition. Similarly,

which together with (3.5)-(3.9) gives

(3.10)

Now, we choose so small that

which together with (3.10) gives (3.4). The proof is complete. □

Lemma 3.4Let (A1)-(A3) hold and, be the solution of (1.1). Then the functionaldefined by

with

satisfies

(3.11)

Proof A direct differentiation for yields

(3.12)

Using the first equation of (1.1) and integrating by parts, we obtain

(3.13)

From Young’s inequality, Poincaré’s inequality and Lemma 3.2, we derive

(3.14)

(3.15)

(3.16)

(3.17)

Now, we estimate the first term on the right-hand side of (3.17). Using the assumptions
(A1)-(A2) and Young’s inequality, we arrive at

(3.18)

where we used the embedding for if or if and the fact proved in Lemma 5.1 in [17]. Combining (3.13)-(3.18), we get

(3.19)

The same estimate to , we can derive

which together with (3.19) gives (3.11). The proof is now complete. □

Proof of Theorem 2.1 For , we define the functional by

and let

for some fixed .

Using Lemma 3.1 and Lemmas 3.3-3.4, a direct differentiation gives

(3.20)

where .

Now, we choose and , large enough so that

(3.21)

(3.22)

(3.23)

Inserting (3.21)-(3.23) into (3.20), we have

(3.24)

Therefore, for two positive constants ω and C, we obtain

(3.25)

On the other hand, we choose even larger so that is equivalent to , i.e.,

(3.26)

Multiplying (3.25) by and using (A3), we get

(3.27)

By virtue of (A3) and , we have

(3.28)

Using (3.26), we can easily get

(3.29)

which together with (3.28) yields, for some positive constant η,

(3.30)

Integrating (3.30) over , we arrive at

which together with (3.29) and the boundedness of E and ξ yields (2.3). The proof is now complete. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The paper is a joint work of all authors who contributed equally to the final version
of the paper. All authors read and approved the final manuscript.

Acknowledgements

Baowei Feng was supported by the Doctoral Innovational Fund of Donghua University
with contract number BC201138, and Yuming Qin was supported by NNSF of China with
contract numbers 11031003 and 11271066 and the grant of Shanghai Education Commission
(No. 13ZZ048).

References

Kafini, M, Tatar, N-e: A decay result to a viscoelastic in with an oscillating kernel. J. Math. Phys.. 51(7), (2010) Article ID 073506